Coherent-state analysis of the seismic head wave problem: an overcomplete representation and its relationship to rays and beams

Abstract

S U M M A R YThe coherent-state transform (CST) is a Gaussian-windowed Fourier transform and compared with the usual plane wave expansion of seismic wavefields it provides an overcomplete basis of partial waves. These overcomplete partial waves have associated rays and the relationship of these rays to those of a physical wave front is explained here by appealing to a familiar analytic example. The exact CST of the standard Fourier plane wave summation for a point-source primary wave can be interpreted as a sum of damped plane waves forming a focussed or beamlike wavefield. This primary wave CST can also be approximated as a bundle of complex rays in a position-slowness space with higher dimension than that usually considered, a consequence of the overcompleteness. The higher-dimensional ray spreading controls the coherent-state (CS) amplitude. This complex-ray bundle in turn can be approximated as a paraxial Gaussian beam carried by a real ray associated with the physical wave front.The exact CST of the standard plane wave summation for a point-source reflection shows how the complex-ray and paraxial-beam approximations generalize to interfaces. Around a critical angle the standard plane wave summation has an asymptotic form involving the Weber function and this function also necessarily arises in the complex-ray and paraxial-beam approximations for the reflection CST. The inverse CST giving the point-source wavefield is a sum of coherent states and those contributing rays or paraxial beams that are incident around the critical angle should be given a reflection coefficient involving the Weber function. CS beams that are incident away from the critical angle collect a peripheral branch-point diffraction. In general, both types of critical-ray/branch-point signal should be included if the integrated CS response is to correctly describe the head wave.An advantage of the CST is that it combines with ray theory for gradually varying media to give a convenient solution to the caustic and pseudocaustic problems. It may be said to unify the Maslov and standard Gaussian-beam methods of seismic modelling, in part by avoiding the ray-centered coordinates conventionally employed in the latter. The general idea of such overlapping partial wave expansions, their flexibility and the smoothing they impart may have other benefits in seismic analysis and processing.

Correlation Migration Using Gaussian Beams of Scattered Teleseismic Body Waves

“…Although a basis cannot be formed by using a windowed Gabor frame, an overcomplete frame expansion can be constructed that has the additional benefit of providing redundancy in the expansion (Feichtinger and Strohmer, 1998). Overcomplete representations of seismic wave fields in terms of coherent states and related Gabor windowed transforms were also described by Thomson (2001), and Thomson (2004) investigated the seismic head wave problem by an overcomplete Gabor representation of beams.…”

Theory and Observations: Body Waves, Ray Methods, and Finite-Frequency Effects

“…Formulations based on the coherent-state transformation technique (Klauder, 1987a,b) fills the gap between local plane wave decomposition of WKBJ/Maslov approaches and overcomplete beam decompositions with a simple parabolic approximation of the phase function (Thomson, 2001(Thomson, , 2004. This allows us to preserve many tracing tools for real rays.…”

Chapter 2. Wavefield Representation, Propagation and Imaging Using Localized Waves: Beamlet, Curvelet and Dreamlet

“…Wavelength being much smaller than the beamwidth and wavefront curvature is the prerequisite of the asymptotic solution. Under the asymptotic regime, different approximations can be applied for the cases of large or small curvature/beamwidth (Klauder, 1987;Thomson, 2001Thomson, , 2004.…”

“…Huang (1991, 2002) apply directly the inverse transform equation (2.32) to synthesize the space domain wavefield, which is a summation of a bundle of asymptotic CS solutions. The other way is apply further a stationary phase approximation (or saddle-point approximation) to the integral and the final solution is similar to a single CS (Klauder, 1987;Thomson, 2001Thomson, , 2004. In this way, the asymptotic CS ray tracing is similar to the classic ray tracing with advantage of avoiding caustics.…”